Draft:Projective Anomaly

In celestial mechanics, Projective anomaly is an angular parameter that defines the position of a body moving along a Keplerian orbit. It is the angle between the direction of periapsis and the current position of the body in the projective space.

The projective anomaly is usually denoted by the $\theta$ and is usually restricted to the range 0 - 360 deg (0 - 2 $\pi$ rad).

The projective anomaly \theta is one of four angular parameters (anomalies) that defines a position along an orbit, the other two being the eccentric anomaly, true anomaly and the mean anomaly.

In the projective geometry, a circle, ellipse, parabola, and hyperbola are treated as the same kind of quadratic curves.

projective parameters and projective anomaly

An orbit type is classified by two project parameters $\alpha$ and $\beta$ as follows,

• circular orbit $\beta =0$

• elliptic orbit $\alpha \beta <1$

• parabolic orbit $\alpha \beta =1$

• hyperbolic orbit $\alpha \beta >1$

• linear orbit $\alpha =\beta$

• imaginary orbit $\alpha <\beta$

where

$\alpha ={\frac {(1+e)(q-p)+{\sqrt {(1+e)^{2}(q+p)^{2}+4e^{2}}}}{2}}$

$\beta ={\frac {2e}{(1+e)(q+p)+{\sqrt {(1+e)^{2}(q+p)^{2}+4e^{2}}}}}$

$q=(1-e)a$

$p={\frac {1}{Q}}={\frac {1}{(1+e)a}}$

where $\alpha$ is semi-major axis,$e$ iseccentricity, $q$ is perihelion distance、$Q$ is aphelion distance.

Position and heliocentric distance of the planet $x$, $y$ and $r$ can be calculated as functions of the projective anomaly $\theta$ :

$x={\frac {-\beta +\alpha \cos \theta }{1+\alpha \beta \cos \theta }}$

$y={\frac {{\sqrt {\alpha ^{2}-\beta ^{2}}}\sin \theta }{1+\alpha \beta \cos \theta }}$

$r={\frac {\alpha -\beta \cos \theta }{1+\alpha \beta \cos \theta }}$

Kelper's equation

The projective anomaly $\theta$ can be calculated from the eccentric anomaly $u$ as follows,

• Case : $\alpha \beta <1$

$\tan {\frac {\theta }{2}}={\sqrt {\frac {1+\alpha \beta }{1-\alpha \beta }}}\tan {\frac {u}{2}}$

$u-e\sin u=M=\left({\frac {1-\alpha ^{2}\beta ^{2}}{\alpha (1+\beta ^{2})}}\right)^{3/2}k(t-T_{0})$

• case : $\alpha \beta =1$

${\frac {s^{3}}{3}}+{\frac {\alpha ^{2}-1}{\alpha ^{2}+1}}s={\frac {2k(t-T_{0})}{\sqrt {\alpha (\alpha ^{2}+1)^{3}}}}$

$s=\tan {\frac {\theta }{2}}$

• case : $\alpha \beta >1$

$\tan {\frac {\theta }{2}}={\sqrt {\frac {\alpha \beta +1}{\alpha \beta -1}}}\tanh {\frac {u}{2}}$

$e\sinh u-u=M=\left({\frac {\alpha ^{2}\beta ^{2}-1}{\alpha (1+\beta ^{2})}}\right)^{3/2}k(t-T_{0})$

The above equations are called Kepler's equation.

Generalized anomaly

For arbitrary constant $\lambda$, thegeneralized anonaly $\Theta$ is related as

$\tan {\frac {\Theta }{2}}=\lambda \tan {\frac {u}{2}}$

The eccentric anomaly, the true anomaly, and the projective anomaly are the cases of $\lambda =1$, $\lambda ={\sqrt {\frac {1+e}{1-e}}}$, $\lambda ={\sqrt {\frac {1+\alpha \beta }{1-\alpha \beta }}}$, respectively.

References

• Sato, I., "A New Anomaly of Keplerian Motion", Astronomical Journal Vol.116, pp.2038-3039, (1997)

See also

• two body problem

• mean anomaly

• eccentric anomaly

• true anomaly

• Kepler's equation

• projective geometry